Compact spaces of separately continuous functions in two variables

Topology and its Applications 107 (2000) 89–96

Compact spaces of separately continuous functions in two variables Sergei P. Gul’ko ∗ , Gennady A. Sokolov Faculty of Mechanics and Mathematics, Tomsk State University, Prospekt Lenina 36/2, Tomsk, 634050, Russia Received 9 November 1998; received in revised form 5 May 1999

Abstract For compacta X and Y , let SCp (X × Y ) be the space of all separately continuous functions on the product X × Y with the topology of pointwise convergence. We prove that any compact subspace Z in SCp (X × Y ) is a Corson compactum in the case where X or Y is ccc (= has the Souslin property). We study this new class of compacta.  2000 Elsevier Science B.V. All rights reserved. Keywords: Separately continuous function; Topology of pointwise convergence; Corson compactum; Eberlein compactum AMS classification: 54C35

1. Introduction It is well known that the important class of Eberlein compact spaces can be characterized as the class of all compacta embeddable into the continuous functions’ space Cp (X) for some compact Hausdorff space X. The question about the structure of compact subsets Z in the space SCp (X × Y ) of separately continuous functions (even in the case of compact factors X and Y ) has an easy but dull answer—in general they may be any compacta. More precisely, the following theorem (having a very simple proof) is true. Theorem 1.1. If the Souslin numbers of both spaces X and Y are not less than a cardinal λ then the Tychonoff cube I λ can be embedded into SCp (X × Y ). However, there is a situation with much more interesting answer to this question. It is related to the case where X or Y has ccc, i.e., the Souslin property. Under this assumption it will be proved here that a compact subspace of SCp (X × Y ) is a Corson compactum. ∗ Corresponding author.

E-mail address: [email protected] (S.P. Gul’ko). 0166-8641/00/$ – see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 8 6 4 1 ( 9 9 ) 0 0 1 1 9 - 4

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The following statements are evident. Lemma 1.2. SCp (X × Y ) is homeomorphic to Cp (X, Cp (Y )) and to Cp (Y, Cp (X)). Lemma 1.3. Let X, Y and Z be compact Hausdorff spaces. Then the following are equivalent: (a) Z can be embedded into SCp (X × Y ). (b) Z can be embedded into Cp (X, Cp (Y )) (or into Cp (Y, Cp (X))). (c) There exists a separately continuous function hx, y, zi on the Cartesian product X × Y × Z which distinguishes points of Z, i.e., z = z0 if and only if hx, y, zi = hx, y, z0 i for any (x, y) ∈ X × Y . Item (c) of this lemma shows that the role of the space Z is different from the roles of X, Y . Let us introduce the following equivalence relation: x ∼ x 0 if and only if hx, y, zi = hx 0 , y, zi for any (y, z) ∈ Y × Z, and denote by X0 the corresponding quotient space. Analogously, we introduce the similar equivalence relation on the space Y and its quotient space Y 0 . Then we get a function on the product X0 × Y 0 × Z which is also separately continuous, but meanwhile it separates points in each space under consideration. If this is true for a given triple of spaces then we will say that they are in a ternary, and the same name we give to the respective function h·, ·, ·i. So, if hX1 , X2 , X3 i is a ternary then the space Xi can be considered as a subspace in SCp (Xj × Xk ) for every i = 1, 2, 3 and j 6= i, k 6= i. We will later often use the described factorization and call it “reduction to ternary”. Note that one of the spaces, namely, the space Z, in the above reasoning remains the same after this reduction. We treat Corollary 2.7 as fundamental, for there some additional reduction to a simpler ternary of spaces is given. Namely, we show that if a space S has ccc and we are interested only of the space X in a ternary hS, X, Y i then it is possible to make the space Y to equal the one-point compactification αΓ of some discrete space Γ . In fact, this statement can be considered as a variant of the Amir and Lindenstrauss theorem, especially in its following formulation: for any duality hX, Y i of compact Hausdorff spaces there exists a new duality hX, αΓ i for some discrete space Γ (of course, the roles of the first and the second spaces in the duality can be changed). The main result of our Corollary 2.7 (for a ccc-space S the sets of compact subspaces in SCp (S × X) and SCp (S × αΓ ) coincide) may be considered as a statement on the existence of some functor from the class of compact Hausdorff spaces with ccc into the class of Corson compact spaces. We only touch this area here, but it seems to us very interesting and promising. The last section contains results permitting us to outline a boundary of a new class of compacta. It is really a new class, because we give examples distinguishing it from very well-known classes of compacta: Corson, Talagrand, Gul’ko. In this paper, we do not affect the problem of characterization of compacta from the new class, and also do not consider “non-compact” ternaries and spaces of functions in more than three variables. Some of these questions will be considered in our forthcoming paper [6].

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Our terminology and notations are standard. The symbol Σ(T ) will denote a usual Σproduct of real lines with the Tychonoff topology.

2. Embedding into a Σ-product The main result of this section is as follows. Theorem 2.1. Let compact Hausdorff spaces S, X, Y be in a ternary and the space S be ccc. Then both X and Y are Corson compacta. Our proof of this theorem follows the well-known method of constructing a “long sequence of projections” (in another terminology, “projectional resolution of identity” (P.R.I.)) due to Amir and Lindenstrauss [1], with a modification given by the first author of this paper (see [3–5,10–12]). This proof is very well known, so we confine ourselves to a sketch of the main points of changes needed. We are going to use the following variant of the construction (cf. [4]). Definition 2.2. Let X be a compact Hausdorff space and λ be an ordinal, λ > ω. The Corson resolution (C.R.) on X is a “long sequence”, that is, a family {pα : ω 6 α 6 λ} of retractions on X such that the following hold: (a) pα ◦ pβ = pβ ◦ pα = pα for α 6 β, (b) pλ is the identity mapping of X, (c) the weight w(pα (X)) 6 |α| for every α, (d) (pointwise continuity) pα (x) = limβ<α pβ (x) for any x ∈ X and for any limit ordinal α, (e) |{α: pα (x) 6= pα+1 (x)| 6 ℵ0 for every x ∈ X. It is easy to see that item (e) of the definition is equivalent to the countable tightness of the space X. In the next lemmas we fix a ternary hS, X, Y i satisfying all conditions of Theorem 2.1. Lemma 2.3. Both compact spaces X and Y are monolithic. Proof. Let A ⊂ X, |A| = λ. Let us consider S as a subset of SCp (X × Y ) and define the mapping fA (s) = s|A×Y . It is continuous and its image S 0 = fA (S) has ccc. For every a ∈ A the space Sa0 = fa (S) ⊂ Cp ({a} × Y ) is a metrizable compactum being an Eberlein Q compact space with ccc [2]. Since S 0 is homeomorphic to the product a∈A Sa0 , we have w(S 0 ) 6 |A| = λ. Next we define the space S 00 in the same way as S 0 , the set A being replaced with its closure A. Evidently, the restriction map fA is a homeomorphism of S 00 onto S 0 . Define an equivalence relation as follows: s1 ∼ s2 if and only if hs1 , x, yi = hs2 , x, yi for any x ∈ A (or x ∈ A, with the same result), y ∈ Y , and let π : S → S 0 (= S 00 ) be the quotient mapping. The formula hπ(s), x, yi = hs, x, yi defines a new ternary hS 00 , A, Y i. Now “invert” the situation and consider the map g : A → SCp (S 00 × Y )

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generated by the last ternary. It is a homeomorphic embedding. Since w(S 00 ) 6 λ, there exists a dense subset B in S 00 of cardinality 6 λ. Then g(A) can be considered as a subset Q of b∈B g(A)|{b}×Y . Therefore, w(g(A)) 6 λ. This proves the monolithic property of X. For the space Y the proof is the same. 2 Lemma 2.4. Let A ⊂ X and B ⊂ Y be sets of cardinality 6 µ, where ω 6 µ < λ. Then there exists a pair of subspaces P ⊂ X and Q ⊂ Y containing A and B, respectively, both having weight 6 µ and satisfying the following conditions: (a) For any x ∈ X there exists p(x) ∈ P such that hs, x, yi = hs, p(x), yi for all y ∈ Q. (b) For any y ∈ Y there exists q(y) ∈ Q such that hs, x, yi = hs, x, q(y)i for all x ∈ P . These two conditions are a slight generalization of our definition of a conjugate pair, first appeared in [5]. We call any two sets P and Q, as in Lemma 2.4, an S-pair. Proof. By the previous lemma, both spaces X and Y are monolithic, and using its proof one can conclude that X|S×B has weight 6 µ, hence, there exists a closed set A1 of weight 6 µ containing A such that X|S×B = A1 |S×B . Then, by inversion of the situation, we find a set B1 , w(B1 ) 6 µ, with Y |S×A1 = B1 |S×A1 , and so on. Using the argument of saturation from [5], we accomplish the construction of required pair P , Q as a result of an inductive process (see [5] or any of the above mentioned references). 2 A straightforward consequence of the last lemma is the existence of a dual pair of retractions in both spaces X and Y . Lemma 2.5. For any S-pair of spaces P , Q there exist retractions p : X → P and q : Y → Q. These retractions are tied by the formula hs, p(x), yi = hs, x, q(y)i which is true for any choice of variables. Proof of Theorem 2.1. One can fix some dense subsets in X and Y , enumerate their points by ordinals and, step by step, construct a “long sequence” of S-pairs (Pα , Qα ), where ω 6 α 6 λ, in such a way that our selected points will be all absorbed at the end of this process. A standard argument shows that this construction satisfies conditions (a)– (d) of Definition 2.2. So we owe to verify condition (e). Suppose this is not true for the space X. Then there is a point x ∈ X such that the set F = {pα (x): ω 6 α < λ} is uncountable. The space F is homeomorphic to some ordinal with the order topology. We may assume that it is a segment of the type [1, ω1 ] (otherwise, we take its closed subspace). Applying the reduction to ternary described in the introduction, we get a new ternary hS 0 , [1, ω1 ], Y 0 i, where the space S 0 has ccc, being a continuous image of S. For the new triple we construct two systems {pα : ω 6 α 6 ω1 } and {qα : ω 6 α 6 ω1 } of retractions in [1, ω1 ] and Y , respectively. Both these systems satisfy properties (a)–(d) of Definition 2.2. From the pointwise continuity it follows that the point ω1 is not contained in pα ([1, ω1 ]) for α < ω1 . This implies the existence of an uncountable closed subset Ω of limit ordinals in [1, ω1 ] such that, for any α ∈ Ω, the mapping pα acts by the formulas: pα (β) = β if β 6 α, and pα (β) = α otherwise. Let us consider the space Y 0 as a compact

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subset of Cp ([1, ω1 ], Cp (S)). If y ∈ Y 0 then y([1, ω1]) is a compact subset of Cp (S). Since S is ccc, y([1, ω1 ]) is a metrizable compact space. It follows that there exists a countable ordinal α(y) such that y(β)(s) is constant for β > α(y) (for any fixed s ∈ S). This fact means that qγ (y, s) = (y, s) for any γ > α(y) and s ∈ S. Therefore, the family {qα (Y × S): α < ω1 }, consisting of metrizable compact spaces, covers the space Y . Let f1 : q1 (Y × S) → I ℵ0 be a homeomorphic embedding into the Hilbert cube and, for any α < ω1 , a mapping fα : qα+1(Y × S)/∼ → I ℵ0 be a homeomorphic embedding of the quotient space qα+1 (Y × S)/∼ obtained by collapsing the set qα (Y × S) into one point ∗ (we denote by hα the quotient mapping). We get the family {f1 ◦ q1 , fα ◦ hα : 1 < α < ω1 } of mappings, which separates points of Y . For any y ∈ Y the set of those α, for which the condition fα ◦ hα (y) 6= ∗ holds, is at most countable. This means that the diagonal product of all these maps embeds Y into a Σ-product of compact metric spaces, hence, it is a Corson compact space. Now we invert the situation and regard the space [1, ω1 ] as consisting of separately continuous functions on the product S × Y . The mapping φ(y) = ω1 (s, y) is continuous and takes its values in Cp (S). Since the space S is ccc, φ(y) is metrizable. So we can find a separable set in Y whose image is equal to the whole set φ(S). Since Y is a Corson compact space, there exists a retraction qα such that φ(y) = φ(qα (y)) for any y ∈ Y (this is a well-known property of Corson compact spaces; also note that any two systems of retractions satisfying (a)–(e) of Definition 2.2 must have uncountable intersection). We may assume also that α ∈ Ω. Writing the last equality with more details, we get hs, ω1 , yi = hs, ω1 , qα (y)i for any s ∈ S and y ∈ Y . Observe that it is possible to choose arbitrarily large α. Repeating the above argument, we find an ordinal β ∈ Ω such that β > α and hs, β, yi = hs, β, qβ (y)i for any s ∈ S, y ∈ Y . Summing all our reasoning, we obtain the following equalities:



 hs, ω1 , yi = s, ω1 , qα (y) = s, ω1 , qβ (y) 



= s, pβ (ω1 ), qβ (y) = s, β, qβ (y) = hs, β, yi, which is true for any s ∈ S and y ∈ Y . Thus, the ternary does not separate the points β and ω1 , a contradiction. 2 Now we give a simple example of a compact space consisting of separately continuous functions which is not an Eberlein compactum. Example 2.6. Suppose that the hypothesis on the existence of a nonseparable Souslin line is valid. Let S be any its nonseparable compact segment. Of course, S has ccc, but it is well known that S × S does not have ccc. Let {Uα : α ∈ ω1 } be an uncountable family of disjoint open subsets of S × S. For every α we define a continuous non-zero function fα with support in Uα . Let Y = {fα : α ∈ ω1 } ∪ {0} where 0 is the zero function. This set is a compact subspace in Cp (S × S) with a single non-isolated point. Since Cp (S × S) ⊂ SCp (S × S), it is possible to use the procedure of reduction to ternary described in the introduction. Then we get a ternary hS 0 , S 00 , Y i where the compact spaces S 0 and S 00 have ccc. By Theorem 2.1, S 0 and S 00 are Corson compact spaces, but both

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of them cannot be Eberlein compacta. For in this case they are metrizable, and the latter immediately forces Y to be metrizable, which is impossible. Corollary 2.7. Let hS, X, Y i be a ternary of compact Hausdorff spaces and S be ccc. Then for some discrete space Γ there exists a new ternary hS, X, αΓ i. Proof. From Lemma 1.2 it follows that there exists an homeomorphic embedding of X into Cp (S, Cp (Y )). Since Y is a Corson compact space, there exists a one-to-one continuous mapping of Cp (Y ) into Cp (αΓ ), by [5]. This implies the existence of a oneto-one continuous mapping of Cp (S, Cp (Y )) into Cp (S, Cp (αΓ )). Hence, the space X is homeomorphically embeddable into this space and, again by Lemma 1.2, into SCp (S × αΓ ). Finally, note that in the new ternary hS, X, αΓ i each space separates points of other spaces. 2 Remark 2.8. If two spaces in a ternary have ccc, for example, the spaces S1 and S2 , then, by Theorem 2.1, every space from the ternary hS1 , S2 , Xi is a Corson compactum. Example 2.6 is just of this type (but X is not metrizable). Remark 2.9. It is evident that if a space S is separable then every compact subspace from SCp (S × αΓ ) is an Eberlein compactum. Denote by SC0p (S × αΓ ) the subspace of SCp (S × αΓ ) consisting of all functions vanishing on the line S × {∞}. Proposition 2.10. The spaces SC0p (S × αΓ ) and SCp (S × αΓ ) are linear homeomorphic. Proof. It is enough to set f → (f |S×{∞} , f (x, y) − f (x, ∞)) to obtain a linear homeomorphism from SCp (S × αΓ ) onto Cp (S × {∞}) × SCp0 (S × αΓ ). The last space is evidently linear homeomorphic to SC0p (S × αΓ ). 2 3. Compact spaces of separately continuous functions Denote by SC the class of all compact spaces lying in SCp (S × X) for some compacta S and X where S has ccc. By Theorem 2.1, it is a variety of Corson compacta and we always can assume X to be equal to αΓ . This class evidently is closed under the operation of taking a closed subspace. Q Theorem 3.1. If Xi is in SCp (Si × αΓi ) for every i ∈ I then the product X = i∈I Xi can L L be identified with a subspace in SCp (S × αΓ ), where S = α( i∈I Si ) and Γ = i∈I Γi . Therefore, we have the following Corollary 3.2. The class SC is closed under countable products.

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The natural question on the invariance of this class under continuous images is still open. Given a set A, we denote by c0 (A) the subspace of RA consisting of all points x with finite ε-support suppε x = {a ∈ A: |x(a)| > ε}. Theorem 3.3. Let X be an element of SC. Then X can be embedded into Σ(T ) in such a way that for any uncountable subset T 0 in T there exists an infinite subset A in T 0 with the property X|A ⊂ c0 (A). Proof. For every γ ∈ Γ we define a function ϕγ : S × X|S×{γ } → R by the formula ϕγ (s, x|γ ) = hs, γ , xi, where x ∈ X, x|γ = x|S×{γ } , γ ∈ Γ . This function is separately continuous on the product of two compact Hausdorff spaces and, by Namioka’s theorem [9], there exists a dense subset Dγ ⊂ S such that ϕγ is continuous at every point of the set Dγ × (X|S×{γ } ). For every γ ∈ Γ , we choose a countable subset Aγ in Dγ such that the restriction mapping X|S×{γ } → X|Aγ ×{γ } is one-to-one (= homeomorphism). This is possible, since the space X|S×{γ } is a metrizable compact space (recall that S has ccc). Further, we γ γ γ enumerate the set Aγ = {s1 , s2 , . . .} arbitrarily and denote E(n) = {(sn , γ ): γ ∈ Γ } and S∞ E = n=1 E(n). Then X|E is homeomorphic to X (E separates points of X), and also Q homeomorphic to the product ∞ n=1 X|E(n) . This fact shows that it is enough to prove our theorem only for the compact space X|E(n) instead of the whole X. So, further we suppose that some n is fixed and X = X|E(n) . Let us fix ε > 0, γ ∈ Γ and s ∈ Aγ . For any x ∈ X|S×{γ } , there exist neighborhoods γ γ γ γ Ux and Vx of the points s and x, respectively such that the oscillation of ϕγ on Ux × Vx is less than ε/2. This is possible by the continuity of ϕγ at the corresponding point. γ γ γ The family {Vx } forms a cover of the compact space X|S×{γ } . Let Vx(1), . . . , Vx(k) be its T γ γ finite subcover, and put Us = ki=1 Ux(i) . This neighborhood have the following property: γ |hs1 , γ , xi − hs2 , γ , xi| < ε/2 for any s1 , s2 ∈ Us and x ∈ X|S×{γ } . γ γ Denote by Tε the family of all pairs (sn , γ ) such that the oscillation of any x ∈ X on Us γ γ

n

is less than ε/2. Suppose that Tε is uncountable. Then the family of open sets Us γ is n uncountable in a space with ccc. Hence, it contains an infinite centered subfamily, say, {Ui }, corresponding to parameters γ = γi , i = 1, 2, . . . . The compactness argument implies the T existence of a point s ∈ ∞ i=1 U i . Then we have hs, γi , xi − hsnγi , γi , xi < ε ∀x ∈ X, i = 1, 2, . . . 2 But the inequality |hs, γi , xi| > ε/2 with fixed s and x, evidently, is true only for a finite number of i’s. Hence, the set  γ γi : |hsn i , γi , xi| > ε is finite for any x ∈ X. We note now that the set Tε has a property which is closely related to the statement of the theorem, but only for the given ε. This defect may be removed very easily. It is enough to take the quotient mapping of R onto itself which collapses the segment [−ε, ε]

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into the point 0, and then to define the corresponding coordinate-wise autoepimorphism gε of Σ(Tε ). It remains to define the required homeomorphism as the diagonal product of the sequence g1/2 , g1/3 , . . . 2 Example 3.4. There are examples of Corson compacta which are not in SC. We present here two of them. The first is Leiderman’s example [7]. Let us consider the family A of P all subsets A in the segment [0, 1] of the real line such that a∈A |b − a| 6 1 for some b ∈ [0, 1]. It is easy to see that any infinite subset B in [0, 1] has an infinite A ∈ A such that A ⊂ B. Let X consist of all characteristic functions χA , where A ∈ A. Then X is an (adequate) compact space and, by Theorem 3.3, it is not in SC. The second example is also an adequate compactum. It first appeared in [8]. Let us fix a subset F of the real line of cardinality ℵ1 and supply it with a well-ordering ≺. Consider the family A of all subsets A ⊂ F on which both orders (≺ and the usual <) either agree or oppose. Then every infinite set in F contains an infinite subset from A. Therefore the space X = {χA : A ∈ A} is a Corson compactum, but is not in the class SC. Example 3.5. The class SC contains a compact space which is not a Gul’ko compactum. Actually, Example 2.6 gives us such a space. Indeed, it is well known [2] that every ccc Gul’ko compact space is metrizable. It would be interesting to find an example of this type, but without any additional set-theoretical assumption. References [1] D. Amir, J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. Math. 88 (1968) 35–46. [2] S. Argyros, S. Negrepontis, On weakly K-countably determined spaces of continuous functions, Proc. Amer. Math. Soc. 87 (1983) 731–736. [3] A.V. Arhangel’ski˘ı, Topological Function Spaces, Kluwer Acad. Publ., Dordrecht, 1992. [4] M.J. Fabian, Gâteaux Differentiability of Convex Functions and Topology, Canadian Math. Soc. Ser., Wiley-Interscience, New York, 1997. [5] S.P. Gul’ko, On the structure of spaces of continuous functions and their complete paracompactness, Russian Math. Surveys 34 (1979) 36–44. [6] S.P. Gul’ko, G.A. Sokolov, Classes of compacta in separately continuous functions’ spaces, in preparation. [7] A.G. Leiderman, On everywhere dense metrizable subspaces of Corson compacta, Math. Notes 38 (3–4) (1985) 751–755. [8] A.G. Leiderman, G.A. Sokolov, Adequate families of sets and Corson compacta, Comment. Math. Univ. Carolin. 25 (1984) 233–246. [9] I. Namioka, Separate continuity and joint continuity, Pacific J. Math. 51 (1974) 515–531. [10] I. Namioka, R.F. Wheeler, Gul’ko’s proof of the Amir–Lindenstrauss theorem, Contemp. Math. 52 (1986) 113–120. [11] S. Negrepontis, Banach spaces and Topology, in: K. Kunen and J.E. Vaughan (Eds.), Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, pp. 1045–1142. [12] R. Pol, On pointwise and weak topology, Preprint 4/84, Warsaw University.